Calculus Examples

$lim_{x \to 8} \frac{4^{x} - 4^{- x}}{4^{x} + 4^{- x}}$

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 4^{x} - 4^{- x}}{lim_{x \to 8} 4^{x} + 4^{- x}}$

Step 2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{lim_{x \to 8} 4^{x} - lim_{x \to 8} 4^{- x}}{lim_{x \to 8} 4^{x} + 4^{- x}}$

Step 3

Move the limit into the exponent.

$\frac{4^{lim_{x \to 8} x} - lim_{x \to 8} 4^{- x}}{lim_{x \to 8} 4^{x} + 4^{- x}}$

Step 4

Move the limit into the exponent.

$\frac{4^{lim_{x \to 8} x} - 4^{lim_{x \to 8} - x}}{lim_{x \to 8} 4^{x} + 4^{- x}}$

Step 5

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{4^{lim_{x \to 8} x} - 4^{- lim_{x \to 8} x}}{lim_{x \to 8} 4^{x} + 4^{- x}}$

Step 6

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $8$ .

$\frac{4^{lim_{x \to 8} x} - 4^{- lim_{x \to 8} x}}{lim_{x \to 8} 4^{x} + lim_{x \to 8} 4^{- x}}$

Step 7

Move the limit into the exponent.

$\frac{4^{lim_{x \to 8} x} - 4^{- lim_{x \to 8} x}}{4^{lim_{x \to 8} x} + lim_{x \to 8} 4^{- x}}$

Step 8

Move the limit into the exponent.

$\frac{4^{lim_{x \to 8} x} - 4^{- lim_{x \to 8} x}}{4^{lim_{x \to 8} x} + 4^{lim_{x \to 8} - x}}$

Step 9

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{4^{lim_{x \to 8} x} - 4^{- lim_{x \to 8} x}}{4^{lim_{x \to 8} x} + 4^{- lim_{x \to 8} x}}$

Step 10

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 10.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{4^{8} - 4^{- lim_{x \to 8} x}}{4^{lim_{x \to 8} x} + 4^{- lim_{x \to 8} x}}$

Step 10.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{4^{8} - 4^{- 8}}{4^{lim_{x \to 8} x} + 4^{- lim_{x \to 8} x}}$

Step 10.3

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{4^{8} - 4^{- 8}}{4^{8} + 4^{- lim_{x \to 8} x}}$

Step 10.4

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{4^{8} - 4^{- 8}}{4^{8} + 4^{- 8}}$

Step 11

Simplify the answer.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Raise $4$ to the power of $8$ .

$\frac{65536 - 4^{- 8}}{4^{8} + 4^{- 8}}$

Step 11.1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{65536 - \frac{1}{4^{8}}}{4^{8} + 4^{- 8}}$

Step 11.1.3

Raise $4$ to the power of $8$ .

$\frac{65536 - \frac{1}{65536}}{4^{8} + 4^{- 8}}$

Step 11.1.4

To write $65536$ as a fraction with a common denominator, multiply by $\frac{65536}{65536}$ .

$\frac{65536 \cdot \frac{65536}{65536} - \frac{1}{65536}}{4^{8} + 4^{- 8}}$

Step 11.1.5

Combine $65536$ and $\frac{65536}{65536}$ .

$\frac{\frac{65536 \cdot 65536}{65536} - \frac{1}{65536}}{4^{8} + 4^{- 8}}$

Step 11.1.6

Combine the numerators over the common denominator.

$\frac{\frac{65536 \cdot 65536 - 1}{65536}}{4^{8} + 4^{- 8}}$

Step 11.1.7

Simplify the numerator.

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Step 11.1.7.1

Multiply $65536$ by $65536$ .

$\frac{\frac{4294967296 - 1}{65536}}{4^{8} + 4^{- 8}}$

Step 11.1.7.2

Subtract $1$ from $4294967296$ .

$\frac{\frac{4294967295}{65536}}{4^{8} + 4^{- 8}}$

Step 11.2

Simplify the denominator.

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Step 11.2.1

Raise $4$ to the power of $8$ .

$\frac{\frac{4294967295}{65536}}{65536 + 4^{- 8}}$

Step 11.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\frac{4294967295}{65536}}{65536 + \frac{1}{4^{8}}}$

Step 11.2.3

Raise $4$ to the power of $8$ .

$\frac{\frac{4294967295}{65536}}{65536 + \frac{1}{65536}}$

Step 11.2.4

To write $65536$ as a fraction with a common denominator, multiply by $\frac{65536}{65536}$ .

$\frac{\frac{4294967295}{65536}}{65536 \cdot \frac{65536}{65536} + \frac{1}{65536}}$

Step 11.2.5

Combine $65536$ and $\frac{65536}{65536}$ .

$\frac{\frac{4294967295}{65536}}{\frac{65536 \cdot 65536}{65536} + \frac{1}{65536}}$

Step 11.2.6

Combine the numerators over the common denominator.

$\frac{\frac{4294967295}{65536}}{\frac{65536 \cdot 65536 + 1}{65536}}$

Step 11.2.7

Simplify the numerator.

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Step 11.2.7.1

Multiply $65536$ by $65536$ .

$\frac{\frac{4294967295}{65536}}{\frac{4294967296 + 1}{65536}}$

Step 11.2.7.2

Add $4294967296$ and $1$ .

$\frac{\frac{4294967295}{65536}}{\frac{4294967297}{65536}}$

Step 11.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{4294967295}{65536} \cdot \frac{65536}{4294967297}$

Step 11.4

Cancel the common factor of $65536$ .

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Step 11.4.1

Cancel the common factor.

$\frac{4294967295}{65536} \cdot \frac{65536}{4294967297}$

Step 11.4.2

Rewrite the expression.

$4294967295 (\frac{1}{4294967297})$

Step 11.5

Combine $4294967295$ and $\frac{1}{4294967297}$ .

$\frac{4294967295}{4294967297}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{4294967295}{4294967297}$

Decimal Form:

$0.99999999 \dots$